60 QUOTA IN PROPORTIONAL REPRESEI^TATION, 



scrutiny has reached the stage at which the number of 

 candidates elected or unexcluded is w + 1. 



16, Using the Hare quota, let us examine what repre- 

 sentation will be obtained by a party as its strength 

 increases from % to 100 %. 



If the strength of the smaller party is less than 100 /2m, 

 i.e., less than half the Hare quota, it cannot obtain a 

 member, however the votes may be distributed among the 

 candidates of the other party. For if the smaller party 

 has only one candidate left, the other party has m, and 

 each of these (the surpluses of any elected candidates hav- 

 ing been distributed) must have more than half the Hare 

 quota; and any other case can be reduced to this. 



If the strength, A, of the smaller party is greater than 

 100/2m and less than 100/w,, the smaller party is. entitled 

 to one m.ember. At the stage under consideration, the 

 smaller party will have only one candidate unexcluded, and 

 his votes will be A . The larger party will have m- — 1 candi- 

 dates, who between them will have 100 — A votes. Conse- 

 quently, if (100 — A)l(m — 1) is greater than A, each 

 of the m — 1 candidates of the larger party may have more 

 votes than the one candidate of the smaller party ; the 

 latter may be excluded, and the larger party will obtain 

 all the seats, instead of m — 1 seats, the number to which 

 it is entitled. The equation — 



A = '"" - f^ (1) 



m — I 



therefore gives the greatest value of A for which this can 

 occur. 



From (1) we get — 



100 



m + 1 



(2) 



The smaller party, then, being entitled to one member, 

 may fail to obtain any representation, if its strength lies 

 between 100/2m and 100/(m + 1) ; so that the range of 

 strength for which disproportionate representation is pos- 

 sible is — 



100 1 100 



m -f 1 2 m 



Similarly, the smaller party, being entitled to two mem- 

 bers, may obtain only one, if its strength lies between 

 3/2 X l60/m and (2m - l)/m x 100/m. In the same 

 way it may be shown that if the strength of the smaller 



